Evaluation of Soil Water Integral Energy Estimation Using Linear and Non-linear Models

Extended Abstract
Introduction
Soil moisture or water content, and the fraction of that water available to plants, are among the most critical aspects of soil water management. The concept of plant-available water was first introduced nearly a century ago by Veihmeyer and Hendrickson (1927), derived from the difference between Field Capacity (FC) and Permanent Wilting Point (PWP). The concepts of the Unrestricted Water Content Range and the Minimum Restricted Water Content Range were proposed by DaSilva et al. (1994). In this framework, in addition to the two moisture limits (FC and PWP), soil aeration and the effect of soil penetration resistance on water availability to the plant are considered using simple relationships. A limitation or defect of the LLWR concept is that it treats the boundary values for aeration porosity, penetration resistance, and soil water potential as abrupt or discontinuous in restricting water availability. In an effort to overcome the shortcomings of these preceding concepts, Minasny and McBratney (2003) proposed the Soil Water Integral Energy (IE) as a criterion for estimating plant-available water in soil, replacing the focus on soil moisture content. Soil water integral energy is a measure of the energy required to extract water from the soil over a specified range of soil water content. Under this concept: firstly, the plant-available water is not solely confined to the PWP and FC range; secondly, the effect of rapid drainage at high moisture contents, which reduces the opportunity for soil water supply to the plant, is taken into account; and thirdly, the limitation imposed by soil hydraulic conductivity at low moisture contents on water flow towards the root and subsequent absorption is incorporated. They utilized various weighting functions across a wide range of soil water potentials, encompassing the potential effect of all limiting physical characteristics on soil water availability. The most significant limitation in employing this index is the time-consuming and costly process of obtaining the soil moisture characteristic curve, the soil penetration resistance curve, and the accuracy or reliability of the coefficients used in defining the proposed weighting functions. Furthermore, in addition to time and expense, errors present in soil sampling and measurement can impose constraints on the application of IE (Integral Energy).
 
Materials and Methods
The study area includes a part of the Tabriz plain. For estimation of IE using the deep learning method, Artificial Neural Network (ANN), and Multiple Linear Regression (MLR), Mathematica Wolfram software version 14.1.0 was utilized. The input features for all three models (MLR, ANN, and Deep ANN) were identical. The data were randomly divided into two groups: training (67 data points) and testing (30 data points). The input features for the models included: 1- Percentage of water-stable aggregates 2- Soil bulk density 3- Porosity 4- Saturated hydraulic conductivity of the soil 5- Percentage of soil texture particles 6- Equivalent calcium carbonate 7- Penetration resistance at saturated moisture 8- Saturated moisture.
 
Results and Discussion
The created models were evaluated using the evaluation statistics of the coefficient of determination R², the adjusted coefficient of determination R²adj, the root mean square error RMSE, the relative error RMSEr, the model efficiency coefficient NSE, and the average percentage of relative error RME.The results showed that the deep learning method with the highest adjusted coefficient of determination (training: 0.998, test: 0.661) and the lowest root mean square error (training: 15.943, test: 118.593), the artificial neural network method (training: 0.945, test: 0.514) and root mean square error (training: 45.347, test: 139.267), and the linear multivariate regression method (training: 0.544, test: 0.317) and root mean square error (training: 126.955, test: 239.264), respectively, provide the best estimate of the IE index.
 
Conclusion
This study underscores the importance of soil water management and the precise assessment of Plant Available Water (PAW). Given the limitations of traditional concepts like PAW and LLWR, particularly their reliance on discontinuous boundaries, the newer the soil water Integral Energy (IE) criterion is adopted as a more accurate measure for estimating plant water availability in soil.The results demonstrated that IE can be effectively and accurately estimated in the studied area (the Tabriz plain) using deep learning techniques (Artificial Neural Networks), relying on a comprehensive set of key soil properties, including the percentage of water-stable aggregates, bulk density, porosity, and saturated hydraulic conductivity. These findings pave the way for applying advanced, data-driven modeling approaches to optimize soil water resource management in arid and semi-arid regions.